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Re: A Geometrical Proof of the Non-invariance of the Spacetime Interval



"RAUBER, JOEL" wrote:

David wrote in response to Hugh:

I'm not denying that you _can_ say that you had breakfast and lunch in
the same place. I'm arguing that it's not valid, geometrically to do so.
When you (the moving observer) determine the spatial distance between
breakfast and lunch at lunchtime, you put the breakfast end of the tape
measure at San Diego, not Los Angeles, and the lunch end of the tape
measure at San Diego. The Los Angeles observers put the breakfast end of
the tape measure at Los Angeles and the lunch end of the tape measure at
San Diego. The locations, in space, of breakfast at the time of lunch,
are what both frames consider to have been the locations of breakfast,
at the time of breakfast, since both frames say that the locations have
not changed, in their frame, since the time of breakfast.

Which strikes me (except for the last sentence below) as a quasi-proof of
John M's statement

<JM> "Clearly then, the spatial distance between two events is a
frame-dependent quantity."

I never claimed that the spatial distance between breakfast and lunch,
in the moving observer's frame, was the same as the spatial distance
between breakfast and lunch, in the stationary observer's frame in SR.
I'm trying to show what the measurements in SR are for this situation.
I'm just saying that they lead to a paradox when you compare the
assigned locations of breakfast, in the two frames, at the time of
lunch, which is when you "draw the lines" between breakfast and lunch,
in the two frames. Each frame draws his line starting from the assigned
location _in space_ of breakfast, at the time of lunch, but he assigns
it _time_ zero, since he says that the location of breakfast at the time
of lunch is the same as it was at the time of breakfast. Therefore, both
lines (from breakfast) start at time zero, but at different spatial
locations. That means that they start at different points in spacetime,
thus, they don't coincide in spacetime and cannot describe a
geometically invariant quantity (the spacetime four-vector between the
events).

These locations don't coincide in space, thus, they don't coincide in
spacetime.


Above you just "proved" that the two events coincide in space for F' but not
for F. So this last statement contradicts your above reasoning.

No, I'm referring to the fact that the assigned locations of the two
events "breakfast", in F and F', don't coincide in space, at the time of
lunch, not that the assigned locations of breakfast and lunch don't
coincide in F'. The assigned location of breakfast in F', at the time of
lunch, (San Diego) doesn't coincide with the assigned location of
breakfast in F (Los Angeles). Remember, the locations of breakfast (for
the purpose of determining the spatial distance between breakfast and
lunch, _and_ determining the "breakfast" endpoint of the line between
breakfast and lunch) are assigned, in each frame, _at the time of
lunch_, since they can't be assigned _before_ the occurrence of lunch.
I'm not saying that the spatial location of breakfast wasn't originally
assigned at the time of breakfast, it was, but the line between
breakfast and lunch, in each frame, begins at the spatial position of
the assigned spatial location of breakfast at the time of lunch (but it
is assigned the time value zero).


Joel R.

--
Dave Rutherford
"New Transformation Equations and the Electric Field Four-vector"
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